It should be noted that the word annulus usually refers to an open annulus.

More generally, one can allow r=0r0r=0 or R=∞RR=\infty. (This makes sense for the purposes of the bound on |z-w|zw|z-w| above.) This would make an annulus include the cases of a punctured disc, and some unbounded domains.

where w∈ℂwℂw\in\mathbb{C}, and rrr and RRR are real numbers with 0<r<R0rR0<r<R.

One can show that two annuli Dw⁢(r,R)subscriptDwrRD_{w}(r,R) and Dw′⁢(r′,R′)subscriptDsuperscriptwnormal-′superscriptrnormal-′superscriptRnormal-′D_{{w^{{\prime}}}}(r^{{\prime}},R^{{\prime}}) are conformally equivalent if and only if R/r=R′/r′RrsuperscriptRnormal-′superscriptrnormal-′R/r=R^{{\prime}}/r^{{\prime}}. More generally, the complement of any closed disk in an open disk is conformally equivalent to precisely one annulus of the form D0⁢(r,1)subscriptD0r1D_{0}(r,1).